Abstract |
Let λ be a complex number in the closed unit disk D , And H be a separable Hilbert space with the orthonormal basis , say ,ε= {e_n:n=0,1,2,…}. A bounded operator T on H is called a λ- Toeplitz operator if <Te_(n+1) ,e_(m+1) >=λ<Te_n ,e_m > (where < , > is inner product on H) The L^2 function φ~ Σa_n e^inθ with a_n=<Te_0 ,e_n> for n>=0 , and a_n=<Te_n ,e_0 > for n<0 is , on the other hand , called the symbol of T The subject arises naturally from a special case of the operator equation S^* AS=λA+B where S is a shift on H , which plays an essential role in finding bounded matrix (a_ij ) on L^2 (Z) that solves the system of equations {((a_(2i,2j) =p_ij+aa_ij@a_(2i,2j-1) =q_ij+ba_ij )@a_(2i-1,2j) =ν_ij+ca_ij@a_(2i-1,2j-1) =ω_ij+da_ij ) ┤, for all i ,j belong Z , where (p_ij ) ,(q_ij ) ,(ν_ij ) ,(ω_ij ) are bounded matrices on l^2 (Z) and a ,b ,c ,d belong C . It is also clear that the well-known Toeplitz operators are precisely the solutions of S^* AS=A , when S is the unilateral shift . In this paper , we will determine the spectra of λ- Toeplitz operators with |λ|=1 of finite order, and when the symbols are analytic with C^1 boundary values. |